Playing with the integral
I have tried numerical integration with different methods and got errors for small vm. Eg. with plain
returns unevaluated, and with the NAG method
N3:=vm->evalf(Int(cosh(v)/sqrt(sech(v)-sech(vm)),v=0..vm,method = _d01ajc));
I get these error messages:
Error, (in evalf/int) unable to obtain a real result
Error, (in evalf/int) NE_QUAD_ROUNDOFF_TOL:
Round-off error prevents the requested tolerance from
being achieved: epsabs = 5.0e-013, epsrel = 5.0e-010.
Also, the plot
shows holes, presumably for intervals where the algorithm fails to converge to a real result.
Curiously, these problems occur where elementary integration is feasible. I wonder the reason of these numerical errors and how to control them. Naively I have increased Digits to 14 and the above plot got much worse!